Rationalisation removes surds from a denominator. The original denominator must be nonzero. (A+B)(A-B)=A2-B2 Choose a multiplier **Denominator** | **Multiply top and bottom by** | **Example** Single surd a b | b | 52 3= 52 3 x 3 3 = 5 36 Binomial a+h b | The conjugate a-h b | 4+ 23- 2= 4+ 23- 2 x 3+ 23+ 2 = 12+4 2+3 2+232-(2)2 = 14+7 27 =2+ 2 Conjugate pairs **Denominator** | **Conjugate** a+h b | a-h b a b+k | a b-k a b+h k | a b-h k Multiply both numerator and denominator by the conjugate. Radicands must be non-negative.

Rationalisation removes surds from a denominator. The original denominator must be nonzero.

\(\begin{aligned} \dfrac{5}{2\sqrt3}&=\dfrac{5}{2\sqrt3}\times\dfrac{\sqrt3}{\sqrt3} \\ &=\dfrac{5\sqrt3}{6} \end{aligned}\)

\(\begin{aligned} \dfrac{4+\sqrt2}{3-\sqrt2}&=\dfrac{4+\sqrt2}{3-\sqrt2}\times\dfrac{3+\sqrt2}{3+\sqrt2} \\ &=\dfrac{12+4\sqrt2+3\sqrt2+2}{3^2-(\sqrt2)^2} \\ &=\dfrac{14+7\sqrt2}{7} \\ &=2+\sqrt2 \end{aligned}\)

Multiply both numerator and denominator by the conjugate. Radicands must be non-negative.

Rationalising the Denominator

Rationalising the Denominator